Share 'Paradox Regarding Random (Normal) Numbers'

I am investigating if some numbers like Pi or SQRT(2) are normal in base 2 or 10, that is, whether any sequence of digits appear with the expected frequency in their decimal representation. Actually, I am even more interested in the nested square root representation (see here) where "digits" are either 0 (with probability 0.43), 1 (with probability 0.30) or 2 (with probability 0.23.) Pretty much…